Theorem fzennn 12767

Description: The cardinality of a finite set of sequential integers. (See om2uz0i 12746 for a description of the hypothesis.) (Contributed by Mario Carneiro, 12-Feb-2013.) (Revised by Mario Carneiro, 7-Mar-2014.)

Hypothesis

Ref	Expression
fzennn.1	|- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om )

Assertion

Ref	Expression
fzennn	|- ( N e. NN0 -> ( 1 ... N ) ~~ ( `' G ` N ) )

Proof of Theorem fzennn

Dummy variables m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . 3 |- ( n = 0 -> ( 1 ... n ) = ( 1 ... 0 ) )
2		fveq2 6191	. . 3 |- ( n = 0 -> ( `' G ` n ) = ( `' G ` 0 ) )
3	1, 2	breq12d 4666	. 2 |- ( n = 0 -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... 0 ) ~~ ( `' G ` 0 ) ) )
4		oveq2 6658	. . 3 |- ( n = m -> ( 1 ... n ) = ( 1 ... m ) )
5		fveq2 6191	. . 3 |- ( n = m -> ( `' G ` n ) = ( `' G ` m ) )
6	4, 5	breq12d 4666	. 2 |- ( n = m -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... m ) ~~ ( `' G ` m ) ) )
7		oveq2 6658	. . 3 |- ( n = ( m + 1 ) -> ( 1 ... n ) = ( 1 ... ( m + 1 ) ) )
8		fveq2 6191	. . 3 |- ( n = ( m + 1 ) -> ( `' G ` n ) = ( `' G ` ( m + 1 ) ) )
9	7, 8	breq12d 4666	. 2 |- ( n = ( m + 1 ) -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... ( m + 1 ) ) ~~ ( `' G ` ( m + 1 ) ) ) )
10		oveq2 6658	. . 3 |- ( n = N -> ( 1 ... n ) = ( 1 ... N ) )
11		fveq2 6191	. . 3 |- ( n = N -> ( `' G ` n ) = ( `' G ` N ) )
12	10, 11	breq12d 4666	. 2 |- ( n = N -> ( ( 1 ... n ) ~~ ( `' G ` n ) <-> ( 1 ... N ) ~~ ( `' G ` N ) ) )
13		0ex 4790	. . . 4 |- (/) e. _V
14	13	enref 7988	. . 3 |- (/) ~~ (/)
15		fz10 12362	. . 3 |- ( 1 ... 0 ) = (/)
16		0z 11388	. . . . . 6 |- 0 e. ZZ
17		fzennn.1	. . . . . 6 |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om )
18	16, 17	om2uzf1oi 12752	. . . . 5 |- G : om -1-1-onto-> ( ZZ>= ` 0 )
19		peano1 7085	. . . . 5 |- (/) e. om
20	18, 19	pm3.2i 471	. . . 4 |- ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ (/) e. om )
21	16, 17	om2uz0i 12746	. . . 4 |- ( G ` (/) ) = 0
22		f1ocnvfv 6534	. . . 4 |- ( ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ (/) e. om ) -> ( ( G ` (/) ) = 0 -> ( `' G ` 0 ) = (/) ) )
23	20, 21, 22	mp2 9	. . 3 |- ( `' G ` 0 ) = (/)
24	14, 15, 23	3brtr4i 4683	. 2 |- ( 1 ... 0 ) ~~ ( `' G ` 0 )
25		simpr 477	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( 1 ... m ) ~~ ( `' G ` m ) )
26		ovex 6678	. . . . . . 7 |- ( m + 1 ) e. _V
27		fvex 6201	. . . . . . 7 |- ( `' G ` m ) e. _V
28		en2sn 8037	. . . . . . 7 |- ( ( ( m + 1 ) e. _V /\ ( `' G ` m ) e. _V ) -> { ( m + 1 ) } ~~ { ( `' G ` m ) } )
29	26, 27, 28	mp2an 708	. . . . . 6 |- { ( m + 1 ) } ~~ { ( `' G ` m ) }
30	29	a1i 11	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> { ( m + 1 ) } ~~ { ( `' G ` m ) } )
31		fzp1disj 12399	. . . . . 6 |- ( ( 1 ... m ) i^i { ( m + 1 ) } ) = (/)
32	31	a1i 11	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( ( 1 ... m ) i^i { ( m + 1 ) } ) = (/) )
33		f1ocnvdm 6540	. . . . . . . . . 10 |- ( ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ m e. ( ZZ>= ` 0 ) ) -> ( `' G ` m ) e. om )
34	18, 33	mpan 706	. . . . . . . . 9 |- ( m e. ( ZZ>= ` 0 ) -> ( `' G ` m ) e. om )
35		nn0uz 11722	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
36	34, 35	eleq2s 2719	. . . . . . . 8 |- ( m e. NN0 -> ( `' G ` m ) e. om )
37		nnord 7073	. . . . . . . 8 |- ( ( `' G ` m ) e. om -> Ord ( `' G ` m ) )
38		ordirr 5741	. . . . . . . 8 |- ( Ord ( `' G ` m ) -> -. ( `' G ` m ) e. ( `' G ` m ) )
39	36, 37, 38	3syl 18	. . . . . . 7 |- ( m e. NN0 -> -. ( `' G ` m ) e. ( `' G ` m ) )
40	39	adantr 481	. . . . . 6 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> -. ( `' G ` m ) e. ( `' G ` m ) )
41		disjsn 4246	. . . . . 6 |- ( ( ( `' G ` m ) i^i { ( `' G ` m ) } ) = (/) <-> -. ( `' G ` m ) e. ( `' G ` m ) )
42	40, 41	sylibr 224	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( ( `' G ` m ) i^i { ( `' G ` m ) } ) = (/) )
43		unen 8040	. . . . 5 |- ( ( ( ( 1 ... m ) ~~ ( `' G ` m ) /\ { ( m + 1 ) } ~~ { ( `' G ` m ) } ) /\ ( ( ( 1 ... m ) i^i { ( m + 1 ) } ) = (/) /\ ( ( `' G ` m ) i^i { ( `' G ` m ) } ) = (/) ) ) -> ( ( 1 ... m ) u. { ( m + 1 ) } ) ~~ ( ( `' G ` m ) u. { ( `' G ` m ) } ) )
44	25, 30, 32, 42, 43	syl22anc 1327	. . . 4 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( ( 1 ... m ) u. { ( m + 1 ) } ) ~~ ( ( `' G ` m ) u. { ( `' G ` m ) } ) )
45		1z 11407	. . . . . 6 |- 1 e. ZZ
46		1m1e0 11089	. . . . . . . . . 10 |- ( 1 - 1 ) = 0
47	46	fveq2i 6194	. . . . . . . . 9 |- ( ZZ>= ` ( 1 - 1 ) ) = ( ZZ>= ` 0 )
48	35, 47	eqtr4i 2647	. . . . . . . 8 |- NN0 = ( ZZ>= ` ( 1 - 1 ) )
49	48	eleq2i 2693	. . . . . . 7 |- ( m e. NN0 <-> m e. ( ZZ>= ` ( 1 - 1 ) ) )
50	49	biimpi 206	. . . . . 6 |- ( m e. NN0 -> m e. ( ZZ>= ` ( 1 - 1 ) ) )
51		fzsuc2 12398	. . . . . 6 |- ( ( 1 e. ZZ /\ m e. ( ZZ>= ` ( 1 - 1 ) ) ) -> ( 1 ... ( m + 1 ) ) = ( ( 1 ... m ) u. { ( m + 1 ) } ) )
52	45, 50, 51	sylancr 695	. . . . 5 |- ( m e. NN0 -> ( 1 ... ( m + 1 ) ) = ( ( 1 ... m ) u. { ( m + 1 ) } ) )
53	52	adantr 481	. . . 4 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( 1 ... ( m + 1 ) ) = ( ( 1 ... m ) u. { ( m + 1 ) } ) )
54		peano2 7086	. . . . . . . . 9 |- ( ( `' G ` m ) e. om -> suc ( `' G ` m ) e. om )
55	36, 54	syl 17	. . . . . . . 8 |- ( m e. NN0 -> suc ( `' G ` m ) e. om )
56	55, 18	jctil 560	. . . . . . 7 |- ( m e. NN0 -> ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ suc ( `' G ` m ) e. om ) )
57	16, 17	om2uzsuci 12747	. . . . . . . . 9 |- ( ( `' G ` m ) e. om -> ( G ` suc ( `' G ` m ) ) = ( ( G ` ( `' G ` m ) ) + 1 ) )
58	36, 57	syl 17	. . . . . . . 8 |- ( m e. NN0 -> ( G ` suc ( `' G ` m ) ) = ( ( G ` ( `' G ` m ) ) + 1 ) )
59	35	eleq2i 2693	. . . . . . . . . . 11 |- ( m e. NN0 <-> m e. ( ZZ>= ` 0 ) )
60	59	biimpi 206	. . . . . . . . . 10 |- ( m e. NN0 -> m e. ( ZZ>= ` 0 ) )
61		f1ocnvfv2 6533	. . . . . . . . . 10 |- ( ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ m e. ( ZZ>= ` 0 ) ) -> ( G ` ( `' G ` m ) ) = m )
62	18, 60, 61	sylancr 695	. . . . . . . . 9 |- ( m e. NN0 -> ( G ` ( `' G ` m ) ) = m )
63	62	oveq1d 6665	. . . . . . . 8 |- ( m e. NN0 -> ( ( G ` ( `' G ` m ) ) + 1 ) = ( m + 1 ) )
64	58, 63	eqtrd 2656	. . . . . . 7 |- ( m e. NN0 -> ( G ` suc ( `' G ` m ) ) = ( m + 1 ) )
65		f1ocnvfv 6534	. . . . . . 7 |- ( ( G : om -1-1-onto-> ( ZZ>= ` 0 ) /\ suc ( `' G ` m ) e. om ) -> ( ( G ` suc ( `' G ` m ) ) = ( m + 1 ) -> ( `' G ` ( m + 1 ) ) = suc ( `' G ` m ) ) )
66	56, 64, 65	sylc 65	. . . . . 6 |- ( m e. NN0 -> ( `' G ` ( m + 1 ) ) = suc ( `' G ` m ) )
67	66	adantr 481	. . . . 5 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( `' G ` ( m + 1 ) ) = suc ( `' G ` m ) )
68		df-suc 5729	. . . . 5 |- suc ( `' G ` m ) = ( ( `' G ` m ) u. { ( `' G ` m ) } )
69	67, 68	syl6eq 2672	. . . 4 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( `' G ` ( m + 1 ) ) = ( ( `' G ` m ) u. { ( `' G ` m ) } ) )
70	44, 53, 69	3brtr4d 4685	. . 3 |- ( ( m e. NN0 /\ ( 1 ... m ) ~~ ( `' G ` m ) ) -> ( 1 ... ( m + 1 ) ) ~~ ( `' G ` ( m + 1 ) ) )
71	70	ex 450	. 2 |- ( m e. NN0 -> ( ( 1 ... m ) ~~ ( `' G ` m ) -> ( 1 ... ( m + 1 ) ) ~~ ( `' G ` ( m + 1 ) ) ) )
72	3, 6, 9, 12, 24, 71	nn0ind 11472	1 |- ( N e. NN0 -> ( 1 ... N ) ~~ ( `' G ` N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   |` cres 5116   Ord word 5722   suc csuc 5725   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   omcom 7065   reccrdg 7505   ~~ cen 7952   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327

This theorem is referenced by:  fzen2  12768  cardfz  12769